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Chapter 6: Problem 95

Find each product. \((x+6)(x-9)\)

Short Answer

Expert verified
The product is \((x^2 - 3x - 54)\).

Step by step solution

01

Apply the Distributive Property

To find the product \((x+6)(x-9)\), use the Distributive Property of multiplication over addition. This property states that each term inside the first parenthesis must be multiplied by each term inside the second parenthesis.
02

Multiply the First Terms

First, multiply the first term of each binomial: \(x \cdot x = x^2\).
03

Multiply the Outer Terms

Next, multiply the outer terms: \(x \cdot (-9) = -9x\).
04

Multiply the Inner Terms

Now, multiply the inner terms: \(6 \cdot x = 6x\).
05

Multiply the Last Terms

Lastly, multiply the last terms: \(6 \cdot (-9) = -54\).
06

Combine Like Terms

Combine all the results from the previous steps: \[ x^2 - 9x + 6x - 54 \]. Next, combine the like terms, \(-9x \) and \(+6x\), to get \(-3x\). The final answer is \[ x^2 - 3x - 54 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The Distributive Property is a foundation of algebra, and it's crucial when multiplying polynomials. This property states that for any numbers or expressions a, b, and c, a(b + c) = ab + ac.
This means you distribute the multiplication over each term inside parentheses.
For our example, (x + 6)(x - 9), we distribute each term in the first parenthesis (x+6) with each term in the second parenthesis (x-9). This gives us four multiplication steps:
Binomial Multiplication
Binomial multiplication involves multiplying two binomials鈥攁 binomial has exactly two terms. To multiply the binomials(x + 6)(x - 9), we need to follow the steps of distributing each term:
  • Multiply the `First` terms:
    x * x = x^2
  • Multiply the `Outer` terms:
    x * (-9) = -9x
  • Multiply the `Inner` terms:
    6 * x = 6x
  • Multiply the `Last` terms:
    6 * (-9) = -54
After multiplying all combinations, you gather all these products together as a sum:
x^2 - 9x + 6x - 54.
Combining Like Terms
Combining like terms is key in simplifying the expression after multiplication. Like terms are terms that have the same variable raised to the same power.
For example, in our expression x^2 - 9x + 6x - 54, both -9x and 6x have the same variable x raised to the first power.
We combine these by adding or subtracting their coefficients:
-9x + 6x = -3x.
Finally, combining all the terms, we have the simplified polynomial:
x^2 - 3x - 54.

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